Quasi-invariants are natural algebraic generalizations of classical invariant polynomials of finite reflection groups. They first appeared in mathematical physics – in the work of O. Chalykh and A. Veselov on quantum integrable systems – in the early 1990s, and since then have found many interesting applications in other areas: most notably, representation theory, algebraic geometry and combinatorics. In this talk, I will explain how the algebras of quasi-invariants arise in topology: as cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result is a generalization of a well-known theorem of A. Borel that realizes the algebra of classical invariant polynomials of a Weyl group W(G) as the cohomology ring of the classifying space BG of the corresponding Lie group G. Perhaps most interesting here is the fact that our construction of spaces of quasi-invariants is purely homotopy-theoretic. It can therefore be extended to some non-Coxeter (p-adic pseudo-reflection) groups, in which case the compact Lie groups are replaced by the so-called p-compact groups (a.k.a. homotopy Lie groups).
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
